---
_id: '40896'
abstract:
- lang: eng
  text: Nonstationary complex random signals are in general improper (not circularly
    symmetric), which means that their complementary covariance is nonzero. Since
    the Karhunen-Loeve (K-L) expansion in its known form is only valid for proper
    processes, we derive the improper version of this expansion. It produces two sets
    of eigenvalues and improper observable coordinates. We then use the K-L expansion
    to solve the problems of detection and estimation of improper complex random signals
    in additive white Gaussian noise. We derive a general result comparing the performance
    of conventional processing, which ignores complementary covariances, with processing
    that takes these into account. In particular, for the detection and estimation
    problems considered, we find that the performance gain, as measured by deflection
    and mean-squared error (MSE), respectively, can be as large as a factor of 2.
    In a communications example, we show how this finding generalizes the result that
    coherent processing enjoys a 3-dB gain over noncoherent processing.
author:
- first_name: Peter J.
  full_name: Schreier, Peter J.
  last_name: Schreier
- first_name: Louis L.
  full_name: Scharf, Louis L.
  last_name: Scharf
- first_name: Clifford T.
  full_name: Mullis, Clifford T.
  last_name: Mullis
citation:
  ama: Schreier PJ, Scharf LL, Mullis CT. Detection and estimation of improper complex
    random signals. <i>IEEE Trans\ Inform\ Theory</i>. 2005;51(1):306–312. doi:<a
    href="https://doi.org/10.1109/TIT.2004.839538">10.1109/TIT.2004.839538</a>
  apa: Schreier, P. J., Scharf, L. L., &#38; Mullis, C. T. (2005). Detection and estimation
    of improper complex random signals. <i>IEEE Trans.\ Inform.\ Theory</i>, <i>51</i>(1),
    306–312. <a href="https://doi.org/10.1109/TIT.2004.839538">https://doi.org/10.1109/TIT.2004.839538</a>
  bibtex: '@article{Schreier_Scharf_Mullis_2005, title={Detection and estimation of
    improper complex random signals}, volume={51}, DOI={<a href="https://doi.org/10.1109/TIT.2004.839538">10.1109/TIT.2004.839538</a>},
    number={1}, journal={IEEE Trans.\ Inform.\ Theory}, author={Schreier, Peter J.
    and Scharf, Louis L. and Mullis, Clifford T.}, year={2005}, pages={306–312} }'
  chicago: 'Schreier, Peter J., Louis L. Scharf, and Clifford T. Mullis. “Detection
    and Estimation of Improper Complex Random Signals.” <i>IEEE Trans.\ Inform.\ Theory</i>
    51, no. 1 (2005): 306–312. <a href="https://doi.org/10.1109/TIT.2004.839538">https://doi.org/10.1109/TIT.2004.839538</a>.'
  ieee: 'P. J. Schreier, L. L. Scharf, and C. T. Mullis, “Detection and estimation
    of improper complex random signals,” <i>IEEE Trans.\ Inform.\ Theory</i>, vol.
    51, no. 1, pp. 306–312, 2005, doi: <a href="https://doi.org/10.1109/TIT.2004.839538">10.1109/TIT.2004.839538</a>.'
  mla: Schreier, Peter J., et al. “Detection and Estimation of Improper Complex Random
    Signals.” <i>IEEE Trans.\ Inform.\ Theory</i>, vol. 51, no. 1, 2005, pp. 306–312,
    doi:<a href="https://doi.org/10.1109/TIT.2004.839538">10.1109/TIT.2004.839538</a>.
  short: P.J. Schreier, L.L. Scharf, C.T. Mullis, IEEE Trans.\ Inform.\ Theory 51
    (2005) 306–312.
date_created: 2023-01-30T11:52:08Z
date_updated: 2023-01-30T11:53:47Z
department:
- _id: '263'
doi: 10.1109/TIT.2004.839538
intvolume: '        51'
issue: '1'
page: 306–312
publication: IEEE Trans.\ Inform.\ Theory
status: public
title: Detection and estimation of improper complex random signals
type: journal_article
user_id: '43497'
volume: 51
year: '2005'
...
---
_id: '40894'
abstract:
- lang: eng
  text: The Rihaczek distribution for stochastic signals is a time- and frequency-shift
    covariant bilinear time-frequency distribution (TFD) based on the Crame acute;r-Loe
    grave;ve spectral representation for a harmonizable process. It is a complex Hilbert
    space inner product (or cross correlation) between the time series and its infinitesimal
    stochastic Fourier generator. To this inner product, we may attach an illuminating
    geometry, wherein the cosine squared of the angle between the time series and
    its infinitesimal stochastic Fourier generator is given by the Rihaczek distribution.
    The Rihaczek distribution also determines a time-varying Wiener filter for estimating
    a time series from its infinitesimal stochastic Fourier generator and measures
    the resulting error covariance. We propose a factored kernel to construct estimators
    of the Rihaczek distribution that are contained in Cohen’s class of bilinear TFDs.
author:
- first_name: Louis L.
  full_name: Scharf, Louis L.
  last_name: Scharf
- first_name: Peter J.
  full_name: Schreier, Peter J.
  last_name: Schreier
- first_name: Alfred
  full_name: Hanssen, Alfred
  last_name: Hanssen
citation:
  ama: Scharf LL, Schreier PJ, Hanssen A. The Hilbert space geometry of the Rihaczek
    distribution for stochastic analytic signals. <i>IEEE Signal Process\ Lett</i>.
    2005;12(4):297–300. doi:<a href="https://doi.org/10.1109/LSP.2005.843772">10.1109/LSP.2005.843772</a>
  apa: Scharf, L. L., Schreier, P. J., &#38; Hanssen, A. (2005). The Hilbert space
    geometry of the Rihaczek distribution for stochastic analytic signals. <i>IEEE
    Signal Process.\ Lett.</i>, <i>12</i>(4), 297–300. <a href="https://doi.org/10.1109/LSP.2005.843772">https://doi.org/10.1109/LSP.2005.843772</a>
  bibtex: '@article{Scharf_Schreier_Hanssen_2005, title={The Hilbert space geometry
    of the Rihaczek distribution for stochastic analytic signals}, volume={12}, DOI={<a
    href="https://doi.org/10.1109/LSP.2005.843772">10.1109/LSP.2005.843772</a>}, number={4},
    journal={IEEE Signal Process.\ Lett.}, author={Scharf, Louis L. and Schreier,
    Peter J. and Hanssen, Alfred}, year={2005}, pages={297–300} }'
  chicago: 'Scharf, Louis L., Peter J. Schreier, and Alfred Hanssen. “The Hilbert
    Space Geometry of the Rihaczek Distribution for Stochastic Analytic Signals.”
    <i>IEEE Signal Process.\ Lett.</i> 12, no. 4 (2005): 297–300. <a href="https://doi.org/10.1109/LSP.2005.843772">https://doi.org/10.1109/LSP.2005.843772</a>.'
  ieee: 'L. L. Scharf, P. J. Schreier, and A. Hanssen, “The Hilbert space geometry
    of the Rihaczek distribution for stochastic analytic signals,” <i>IEEE Signal
    Process.\ Lett.</i>, vol. 12, no. 4, pp. 297–300, 2005, doi: <a href="https://doi.org/10.1109/LSP.2005.843772">10.1109/LSP.2005.843772</a>.'
  mla: Scharf, Louis L., et al. “The Hilbert Space Geometry of the Rihaczek Distribution
    for Stochastic Analytic Signals.” <i>IEEE Signal Process.\ Lett.</i>, vol. 12,
    no. 4, 2005, pp. 297–300, doi:<a href="https://doi.org/10.1109/LSP.2005.843772">10.1109/LSP.2005.843772</a>.
  short: L.L. Scharf, P.J. Schreier, A. Hanssen, IEEE Signal Process.\ Lett. 12 (2005)
    297–300.
date_created: 2023-01-30T11:52:08Z
date_updated: 2023-01-30T11:53:06Z
department:
- _id: '263'
doi: 10.1109/LSP.2005.843772
intvolume: '        12'
issue: '4'
page: 297–300
publication: IEEE Signal Process.\ Lett.
status: public
title: The Hilbert space geometry of the Rihaczek distribution for stochastic analytic
  signals
type: journal_article
user_id: '43497'
volume: 12
year: '2005'
...
---
_id: '40893'
abstract:
- lang: eng
  text: Based on the Cramer-Loeve spectral representation for a harmonizable random
    process, the Rihaczek distribution is a time- and frequency-shift covariant, bilinear
    time-frequency distribution. It can be expressed as a complex Hilbert space inner
    product between the time series and its infinitesimal stochastic Fourier generator.
    We show that we may attach an illuminating geometry to this inner product, wherein
    the cosine-squared of the angle between the time series and its infinitesimal
    stochastic Fourier generator is given by the Rihaczek distribution. We propose
    to construct estimators of the Rihaczek distribution using a factored kernel in
    Cohen’s class of bilinear time-frequency distributions
author:
- first_name: Peter J.
  full_name: Schreier, Peter J.
  last_name: Schreier
- first_name: Louis L.
  full_name: Scharf, Louis L.
  last_name: Scharf
- first_name: Alfred
  full_name: Hanssen, Alfred
  last_name: Hanssen
citation:
  ama: 'Schreier PJ, Scharf LL, Hanssen A. A geometric interpretation of the Rihaczek
    time-frequency distribution for stochastic signals. In: <i>Proc.\ IEEE Int.\ Symp.\
    Inform.\ Theory</i>. ; 2005:966–969. doi:<a href="https://doi.org/10.1109/ISIT.2005.1523481">10.1109/ISIT.2005.1523481</a>'
  apa: Schreier, P. J., Scharf, L. L., &#38; Hanssen, A. (2005). A geometric interpretation
    of the Rihaczek time-frequency distribution for stochastic signals. <i>Proc.\
    IEEE Int.\ Symp.\ Inform.\ Theory</i>, 966–969. <a href="https://doi.org/10.1109/ISIT.2005.1523481">https://doi.org/10.1109/ISIT.2005.1523481</a>
  bibtex: '@inproceedings{Schreier_Scharf_Hanssen_2005, title={A geometric interpretation
    of the Rihaczek time-frequency distribution for stochastic signals}, DOI={<a href="https://doi.org/10.1109/ISIT.2005.1523481">10.1109/ISIT.2005.1523481</a>},
    booktitle={Proc.\ IEEE Int.\ Symp.\ Inform.\ Theory}, author={Schreier, Peter
    J. and Scharf, Louis L. and Hanssen, Alfred}, year={2005}, pages={966–969} }'
  chicago: Schreier, Peter J., Louis L. Scharf, and Alfred Hanssen. “A Geometric Interpretation
    of the Rihaczek Time-Frequency Distribution for Stochastic Signals.” In <i>Proc.\
    IEEE Int.\ Symp.\ Inform.\ Theory</i>, 966–969, 2005. <a href="https://doi.org/10.1109/ISIT.2005.1523481">https://doi.org/10.1109/ISIT.2005.1523481</a>.
  ieee: 'P. J. Schreier, L. L. Scharf, and A. Hanssen, “A geometric interpretation
    of the Rihaczek time-frequency distribution for stochastic signals,” in <i>Proc.\
    IEEE Int.\ Symp.\ Inform.\ Theory</i>, 2005, pp. 966–969, doi: <a href="https://doi.org/10.1109/ISIT.2005.1523481">10.1109/ISIT.2005.1523481</a>.'
  mla: Schreier, Peter J., et al. “A Geometric Interpretation of the Rihaczek Time-Frequency
    Distribution for Stochastic Signals.” <i>Proc.\ IEEE Int.\ Symp.\ Inform.\ Theory</i>,
    2005, pp. 966–969, doi:<a href="https://doi.org/10.1109/ISIT.2005.1523481">10.1109/ISIT.2005.1523481</a>.
  short: 'P.J. Schreier, L.L. Scharf, A. Hanssen, in: Proc.\ IEEE Int.\ Symp.\ Inform.\
    Theory, 2005, pp. 966–969.'
date_created: 2023-01-30T11:52:08Z
date_updated: 2023-01-30T11:53:44Z
department:
- _id: '263'
doi: 10.1109/ISIT.2005.1523481
page: 966–969
publication: Proc.\ IEEE Int.\ Symp.\ Inform.\ Theory
status: public
title: A geometric interpretation of the Rihaczek time-frequency distribution for
  stochastic signals
type: conference
user_id: '43497'
year: '2005'
...
---
_id: '40895'
abstract:
- lang: eng
  text: 'There are two types of aliasing in higher order spectra: “regular aliasing”
    due to sampling below the Nyquist frequency, and “higher order aliasing”. Spectra
    of discrete-time signals may suffer from higher-order aliasing if the signals
    are not sufficiently oversampled. By providing some insight into the cause of
    higher order aliasing, we show that higher order aliasing can just as well occur
    in second order spectra. More importantly, we demonstrate that spectra of stationary
    random signals defined as ensemble-averages and spectra of ergodic random signals
    defined as the Fourier transform of infinite time-averages never exhibit higher
    order aliasing'
author:
- first_name: Peter J.
  full_name: Schreier, Peter J.
  last_name: Schreier
citation:
  ama: 'Schreier PJ. A note on aliasing in higher order spectra. In: <i>Proc. 6th\
    Australian Comm.\ Theory Works.</i> ; 2005:184–188. doi:<a href="https://doi.org/10.1109/AUSCTW.2005.1624249">10.1109/AUSCTW.2005.1624249</a>'
  apa: Schreier, P. J. (2005). A note on aliasing in higher order spectra. <i>Proc.
    6th\ Australian Comm.\ Theory Works.</i>, 184–188. <a href="https://doi.org/10.1109/AUSCTW.2005.1624249">https://doi.org/10.1109/AUSCTW.2005.1624249</a>
  bibtex: '@inproceedings{Schreier_2005, title={A note on aliasing in higher order
    spectra}, DOI={<a href="https://doi.org/10.1109/AUSCTW.2005.1624249">10.1109/AUSCTW.2005.1624249</a>},
    booktitle={Proc. 6th\ Australian Comm.\ Theory Works.}, author={Schreier, Peter
    J.}, year={2005}, pages={184–188} }'
  chicago: Schreier, Peter J. “A Note on Aliasing in Higher Order Spectra.” In <i>Proc.
    6th\ Australian Comm.\ Theory Works.</i>, 184–188, 2005. <a href="https://doi.org/10.1109/AUSCTW.2005.1624249">https://doi.org/10.1109/AUSCTW.2005.1624249</a>.
  ieee: 'P. J. Schreier, “A note on aliasing in higher order spectra,” in <i>Proc.
    6th\ Australian Comm.\ Theory Works.</i>, 2005, pp. 184–188, doi: <a href="https://doi.org/10.1109/AUSCTW.2005.1624249">10.1109/AUSCTW.2005.1624249</a>.'
  mla: Schreier, Peter J. “A Note on Aliasing in Higher Order Spectra.” <i>Proc. 6th\
    Australian Comm.\ Theory Works.</i>, 2005, pp. 184–188, doi:<a href="https://doi.org/10.1109/AUSCTW.2005.1624249">10.1109/AUSCTW.2005.1624249</a>.
  short: 'P.J. Schreier, in: Proc. 6th\ Australian Comm.\ Theory Works., 2005, pp.
    184–188.'
date_created: 2023-01-30T11:52:08Z
date_updated: 2023-01-30T11:53:51Z
department:
- _id: '263'
doi: 10.1109/AUSCTW.2005.1624249
page: 184–188
publication: Proc. 6th\ Australian Comm.\ Theory Works.
status: public
title: A note on aliasing in higher order spectra
type: conference
user_id: '43497'
year: '2005'
...
---
_id: '40897'
abstract:
- lang: eng
  text: The tracking of the time-varying channel is crucial for MIMO-OFDM wireless
    communication systems. Aiming at doubly-selective fading MIMO channels, we present
    a novel pilot pattern and the corresponding adaptive channel tracking algorithm.
    Compared with the traditional pilot patterns, the new pattern has higher frequency
    efficiency and is more suitable for large transmit arrays. The computational complexity,
    frequency efficiency and BER performance of the system assisted by the proposed
    channel tracking scheme are analyzed. Simulation results demonstrate that the
    proposed scheme can track the timevarying channel effectively.
author:
- first_name: Jun
  full_name: Tong, Jun
  last_name: Tong
- first_name: Yaohuan
  full_name: Gong, Yaohuan
  last_name: Gong
- first_name: Shengxian
  full_name: Sun, Shengxian
  last_name: Sun
citation:
  ama: 'Tong J, Gong Y, Sun S. An adaptive channel tracking method for MIMO-OFDM systems.
    In: <i>Int.\ Conf.\ Comm.\ Circuits Syst.</i> Vol 1. ; 2004:354–358. doi:<a href="https://doi.org/10.1109/ICCCAS.2004.1346098">10.1109/ICCCAS.2004.1346098</a>'
  apa: Tong, J., Gong, Y., &#38; Sun, S. (2004). An adaptive channel tracking method
    for MIMO-OFDM systems. <i>Int.\ Conf.\ Comm.\ Circuits Syst.</i>, <i>1</i>, 354–358.
    <a href="https://doi.org/10.1109/ICCCAS.2004.1346098">https://doi.org/10.1109/ICCCAS.2004.1346098</a>
  bibtex: '@inproceedings{Tong_Gong_Sun_2004, title={An adaptive channel tracking
    method for MIMO-OFDM systems}, volume={1}, DOI={<a href="https://doi.org/10.1109/ICCCAS.2004.1346098">10.1109/ICCCAS.2004.1346098</a>},
    booktitle={Int.\ Conf.\ Comm.\ Circuits Syst.}, author={Tong, Jun and Gong, Yaohuan
    and Sun, Shengxian}, year={2004}, pages={354–358} }'
  chicago: Tong, Jun, Yaohuan Gong, and Shengxian Sun. “An Adaptive Channel Tracking
    Method for MIMO-OFDM Systems.” In <i>Int.\ Conf.\ Comm.\ Circuits Syst.</i>, 1:354–358,
    2004. <a href="https://doi.org/10.1109/ICCCAS.2004.1346098">https://doi.org/10.1109/ICCCAS.2004.1346098</a>.
  ieee: 'J. Tong, Y. Gong, and S. Sun, “An adaptive channel tracking method for MIMO-OFDM
    systems,” in <i>Int.\ Conf.\ Comm.\ Circuits Syst.</i>, 2004, vol. 1, pp. 354–358,
    doi: <a href="https://doi.org/10.1109/ICCCAS.2004.1346098">10.1109/ICCCAS.2004.1346098</a>.'
  mla: Tong, Jun, et al. “An Adaptive Channel Tracking Method for MIMO-OFDM Systems.”
    <i>Int.\ Conf.\ Comm.\ Circuits Syst.</i>, vol. 1, 2004, pp. 354–358, doi:<a href="https://doi.org/10.1109/ICCCAS.2004.1346098">10.1109/ICCCAS.2004.1346098</a>.
  short: 'J. Tong, Y. Gong, S. Sun, in: Int.\ Conf.\ Comm.\ Circuits Syst., 2004,
    pp. 354–358.'
date_created: 2023-01-30T11:52:08Z
date_updated: 2023-01-30T11:53:54Z
department:
- _id: '263'
doi: 10.1109/ICCCAS.2004.1346098
intvolume: '         1'
page: 354–358
publication: Int.\ Conf.\ Comm.\ Circuits Syst.
status: public
title: An adaptive channel tracking method for MIMO-OFDM systems
type: conference
user_id: '43497'
volume: 1
year: '2004'
...
---
_id: '40898'
abstract:
- lang: eng
  text: For complex signals, n-th order moment functions can be defined in 2n different
    ways, depending on the placement of complex conjugates. We demonstrate that, for
    stationary analytic signals, only a few of these different moments are actually
    required for a complete n-th order description. Which, and how many of them, depends
    on the signal’s spectrum. We investigate properties of n-th order moments and
    spectra with different conjugation patterns and show how they provide different
    information about the signal.
author:
- first_name: Peter J.
  full_name: Schreier, Peter J.
  last_name: Schreier
- first_name: Louis L.
  full_name: Scharf, Louis L.
  last_name: Scharf
citation:
  ama: 'Schreier PJ, Scharf LL. Polyspectra of analytic signals. In: <i>Proc.\ IEEE
    Int.\ Conf.\ Acoustics, Speech and Signal Process.</i> Vol 2. ; 2004:473–476.
    doi:<a href="https://doi.org/10.1109/ICASSP.2004.1326297">10.1109/ICASSP.2004.1326297</a>'
  apa: Schreier, P. J., &#38; Scharf, L. L. (2004). Polyspectra of analytic signals.
    <i>Proc.\ IEEE Int.\ Conf.\ Acoustics, Speech and Signal Process.</i>, <i>2</i>,
    473–476. <a href="https://doi.org/10.1109/ICASSP.2004.1326297">https://doi.org/10.1109/ICASSP.2004.1326297</a>
  bibtex: '@inproceedings{Schreier_Scharf_2004, title={Polyspectra of analytic signals},
    volume={2}, DOI={<a href="https://doi.org/10.1109/ICASSP.2004.1326297">10.1109/ICASSP.2004.1326297</a>},
    booktitle={Proc.\ IEEE Int.\ Conf.\ Acoustics, Speech and Signal Process.}, author={Schreier,
    Peter J. and Scharf, Louis L.}, year={2004}, pages={473–476} }'
  chicago: Schreier, Peter J., and Louis L. Scharf. “Polyspectra of Analytic Signals.”
    In <i>Proc.\ IEEE Int.\ Conf.\ Acoustics, Speech and Signal Process.</i>, 2:473–476,
    2004. <a href="https://doi.org/10.1109/ICASSP.2004.1326297">https://doi.org/10.1109/ICASSP.2004.1326297</a>.
  ieee: 'P. J. Schreier and L. L. Scharf, “Polyspectra of analytic signals,” in <i>Proc.\
    IEEE Int.\ Conf.\ Acoustics, Speech and Signal Process.</i>, 2004, vol. 2, pp.
    473–476, doi: <a href="https://doi.org/10.1109/ICASSP.2004.1326297">10.1109/ICASSP.2004.1326297</a>.'
  mla: Schreier, Peter J., and Louis L. Scharf. “Polyspectra of Analytic Signals.”
    <i>Proc.\ IEEE Int.\ Conf.\ Acoustics, Speech and Signal Process.</i>, vol. 2,
    2004, pp. 473–476, doi:<a href="https://doi.org/10.1109/ICASSP.2004.1326297">10.1109/ICASSP.2004.1326297</a>.
  short: 'P.J. Schreier, L.L. Scharf, in: Proc.\ IEEE Int.\ Conf.\ Acoustics, Speech
    and Signal Process., 2004, pp. 473–476.'
date_created: 2023-01-30T11:52:08Z
date_updated: 2023-01-30T11:53:59Z
department:
- _id: '263'
doi: 10.1109/ICASSP.2004.1326297
intvolume: '         2'
page: 473–476
publication: Proc.\ IEEE Int.\ Conf.\ Acoustics, Speech and Signal Process.
status: public
title: Polyspectra of analytic signals
type: conference
user_id: '43497'
volume: 2
year: '2004'
...
---
_id: '40903'
abstract:
- lang: eng
  text: Non-stationary complex random signals are in general improper (not circularly
    symmetric), which means that their complementary covariance is non-zero. Since
    the Karhunen-Loeve expansion in its known form is only valid for proper processes,
    we derive the improper version of this expansion. It produces two sets of eigenvalues
    and an improper internal description. We use the Karhunen-Loeve expansion to solve
    the problem of detecting non-stationary improper complex random signals in additive
    white Gaussian noise. Using the deflection criterion we compare the performance
    of conventional processing, which ignores complementary covariances, with processing
    that takes these into account. The performance gain can be as great as a factor
    of 2.
author:
- first_name: Peter J.
  full_name: Schreier, Peter J.
  last_name: Schreier
- first_name: Louis L.
  full_name: Scharf, Louis L.
  last_name: Scharf
citation:
  ama: 'Schreier PJ, Scharf LL. The Karhunen-Loève expansion of improper complex random
    signals with applications in detection. In: <i>Proc.\ IEEE Int.\ Conf.\ Acoustics,
    Speech and Signal Process.</i> Vol 6. ; 2003:717–720. doi:<a href="https://doi.org/10.1109/ICASSP.2003.1201782">10.1109/ICASSP.2003.1201782</a>'
  apa: Schreier, P. J., &#38; Scharf, L. L. (2003). The Karhunen-Loève expansion of
    improper complex random signals with applications in detection. <i>Proc.\ IEEE
    Int.\ Conf.\ Acoustics, Speech and Signal Process.</i>, <i>6</i>, 717–720. <a
    href="https://doi.org/10.1109/ICASSP.2003.1201782">https://doi.org/10.1109/ICASSP.2003.1201782</a>
  bibtex: '@inproceedings{Schreier_Scharf_2003, title={The Karhunen-Loève expansion
    of improper complex random signals with applications in detection}, volume={6},
    DOI={<a href="https://doi.org/10.1109/ICASSP.2003.1201782">10.1109/ICASSP.2003.1201782</a>},
    booktitle={Proc.\ IEEE Int.\ Conf.\ Acoustics, Speech and Signal Process.}, author={Schreier,
    Peter J. and Scharf, Louis L.}, year={2003}, pages={717–720} }'
  chicago: Schreier, Peter J., and Louis L. Scharf. “The Karhunen-Loève Expansion
    of Improper Complex Random Signals with Applications in Detection.” In <i>Proc.\
    IEEE Int.\ Conf.\ Acoustics, Speech and Signal Process.</i>, 6:717–720, 2003.
    <a href="https://doi.org/10.1109/ICASSP.2003.1201782">https://doi.org/10.1109/ICASSP.2003.1201782</a>.
  ieee: 'P. J. Schreier and L. L. Scharf, “The Karhunen-Loève expansion of improper
    complex random signals with applications in detection,” in <i>Proc.\ IEEE Int.\
    Conf.\ Acoustics, Speech and Signal Process.</i>, 2003, vol. 6, pp. 717–720, doi:
    <a href="https://doi.org/10.1109/ICASSP.2003.1201782">10.1109/ICASSP.2003.1201782</a>.'
  mla: Schreier, Peter J., and Louis L. Scharf. “The Karhunen-Loève Expansion of Improper
    Complex Random Signals with Applications in Detection.” <i>Proc.\ IEEE Int.\ Conf.\
    Acoustics, Speech and Signal Process.</i>, vol. 6, 2003, pp. 717–720, doi:<a href="https://doi.org/10.1109/ICASSP.2003.1201782">10.1109/ICASSP.2003.1201782</a>.
  short: 'P.J. Schreier, L.L. Scharf, in: Proc.\ IEEE Int.\ Conf.\ Acoustics, Speech
    and Signal Process., 2003, pp. 717–720.'
date_created: 2023-01-30T11:52:09Z
date_updated: 2023-01-30T11:52:42Z
department:
- _id: '263'
doi: 10.1109/ICASSP.2003.1201782
intvolume: '         6'
page: 717–720
publication: Proc.\ IEEE Int.\ Conf.\ Acoustics, Speech and Signal Process.
status: public
title: The Karhunen-Loève expansion of improper complex random signals with applications
  in detection
type: conference
user_id: '43497'
volume: 6
year: '2003'
...
---
_id: '40900'
abstract:
- lang: eng
  text: In this paper we describe a beamforming algorithm based on widely-linear rather
    than linear data models. Initially, we develop this beamformer by generalizing
    the Capon (MVDR) optimization problem. That is, if the objective is to minimize
    output power while maintaining a specified directional gain, then we show that
    the output power of the widely-linear beamformer is less than or equal to the
    output power of the Capon (MVDR) beamformer. This result is valid regardless of
    the “true” distribution of the data. We also derive the widely-linear beamformer
    by considering beamforming to be an estimation problem. Linear models assume that
    the composite covariance matrix formed from the real and imaginary parts of the
    array-snapshot has a particular structure. This structure is often summarized
    by stating that the covariance formed from the array snapshot and its transpose
    (not Hermitian transpose) is zero. We could also call these data “proper” Gaussian
    vectors. The beamformers in this paper are appropriate for situations in which
    these implicit assumptions are violated.
author:
- first_name: Todd
  full_name: McWhorter, Todd
  last_name: McWhorter
- first_name: Peter J.
  full_name: Schreier, Peter J.
  last_name: Schreier
citation:
  ama: 'McWhorter T, Schreier PJ. Widely-linear beamforming. In: <i>Proc. 37th\ Asilomar
    Conf.\ Signals Syst.\ Computers</i>. Vol 1. ; 2003:753–759. doi:<a href="https://doi.org/10.1109/ACSSC.2003.1292015">10.1109/ACSSC.2003.1292015</a>'
  apa: McWhorter, T., &#38; Schreier, P. J. (2003). Widely-linear beamforming. <i>Proc.
    37th\ Asilomar Conf.\ Signals Syst.\ Computers</i>, <i>1</i>, 753–759. <a href="https://doi.org/10.1109/ACSSC.2003.1292015">https://doi.org/10.1109/ACSSC.2003.1292015</a>
  bibtex: '@inproceedings{McWhorter_Schreier_2003, title={Widely-linear beamforming},
    volume={1}, DOI={<a href="https://doi.org/10.1109/ACSSC.2003.1292015">10.1109/ACSSC.2003.1292015</a>},
    booktitle={Proc. 37th\ Asilomar Conf.\ Signals Syst.\ Computers}, author={McWhorter,
    Todd and Schreier, Peter J.}, year={2003}, pages={753–759} }'
  chicago: McWhorter, Todd, and Peter J. Schreier. “Widely-Linear Beamforming.” In
    <i>Proc. 37th\ Asilomar Conf.\ Signals Syst.\ Computers</i>, 1:753–759, 2003.
    <a href="https://doi.org/10.1109/ACSSC.2003.1292015">https://doi.org/10.1109/ACSSC.2003.1292015</a>.
  ieee: 'T. McWhorter and P. J. Schreier, “Widely-linear beamforming,” in <i>Proc.
    37th\ Asilomar Conf.\ Signals Syst.\ Computers</i>, 2003, vol. 1, pp. 753–759,
    doi: <a href="https://doi.org/10.1109/ACSSC.2003.1292015">10.1109/ACSSC.2003.1292015</a>.'
  mla: McWhorter, Todd, and Peter J. Schreier. “Widely-Linear Beamforming.” <i>Proc.
    37th\ Asilomar Conf.\ Signals Syst.\ Computers</i>, vol. 1, 2003, pp. 753–759,
    doi:<a href="https://doi.org/10.1109/ACSSC.2003.1292015">10.1109/ACSSC.2003.1292015</a>.
  short: 'T. McWhorter, P.J. Schreier, in: Proc. 37th\ Asilomar Conf.\ Signals Syst.\
    Computers, 2003, pp. 753–759.'
date_created: 2023-01-30T11:52:09Z
date_updated: 2023-01-30T11:52:45Z
department:
- _id: '263'
doi: 10.1109/ACSSC.2003.1292015
intvolume: '         1'
page: 753–759
publication: Proc. 37th\ Asilomar Conf.\ Signals Syst.\ Computers
status: public
title: Widely-linear beamforming
type: conference
user_id: '43497'
volume: 1
year: '2003'
...
---
_id: '40901'
abstract:
- lang: eng
  text: Historically, transform coding of noisy sources has been performed by first
    estimating the message and then quantizing this estimate. We show that it is also
    optimum to first transform the noisy observations into canonical coordinates,
    quantize, apply a Wiener filter in this coordinate system, and then transform
    the result back to the original coordinates. Canonical coordinates are uncorrelated,
    and quantizing and Wiener filtering are applied to each component independently.
    Optimality of this approach can be proved assuming additive white quantization
    noise. Half canonical coordinates minimize the mean-squared error by minimizing
    the trace of the error covariance matrix and full canonical coordinates maximize
    information rate by minimizing the determinant of the error covariance matrix.
author:
- first_name: Peter J.
  full_name: Schreier, Peter J.
  last_name: Schreier
- first_name: Louis L.
  full_name: Scharf, Louis L.
  last_name: Scharf
- first_name: Tianjian
  full_name: Hu, Tianjian
  last_name: Hu
- first_name: Stephen D.
  full_name: Voran, Stephen D.
  last_name: Voran
citation:
  ama: 'Schreier PJ, Scharf LL, Hu T, Voran SD. Canonical coordinates are the right
    coordinate system for transform coding of noisy sources. In: <i>Proc.\ IEEE Works.\
    Statistical Signal Proces.</i> ; 2003:234–237. doi:<a href="https://doi.org/10.1109/SSP.2003.1289387">10.1109/SSP.2003.1289387</a>'
  apa: Schreier, P. J., Scharf, L. L., Hu, T., &#38; Voran, S. D. (2003). Canonical
    coordinates are the right coordinate system for transform coding of noisy sources.
    <i>Proc.\ IEEE Works.\ Statistical Signal Proces.</i>, 234–237. <a href="https://doi.org/10.1109/SSP.2003.1289387">https://doi.org/10.1109/SSP.2003.1289387</a>
  bibtex: '@inproceedings{Schreier_Scharf_Hu_Voran_2003, title={Canonical coordinates
    are the right coordinate system for transform coding of noisy sources}, DOI={<a
    href="https://doi.org/10.1109/SSP.2003.1289387">10.1109/SSP.2003.1289387</a>},
    booktitle={Proc.\ IEEE Works.\ Statistical Signal Proces.}, author={Schreier,
    Peter J. and Scharf, Louis L. and Hu, Tianjian and Voran, Stephen D.}, year={2003},
    pages={234–237} }'
  chicago: Schreier, Peter J., Louis L. Scharf, Tianjian Hu, and Stephen D. Voran.
    “Canonical Coordinates Are the Right Coordinate System for Transform Coding of
    Noisy Sources.” In <i>Proc.\ IEEE Works.\ Statistical Signal Proces.</i>, 234–237,
    2003. <a href="https://doi.org/10.1109/SSP.2003.1289387">https://doi.org/10.1109/SSP.2003.1289387</a>.
  ieee: 'P. J. Schreier, L. L. Scharf, T. Hu, and S. D. Voran, “Canonical coordinates
    are the right coordinate system for transform coding of noisy sources,” in <i>Proc.\
    IEEE Works.\ Statistical Signal Proces.</i>, 2003, pp. 234–237, doi: <a href="https://doi.org/10.1109/SSP.2003.1289387">10.1109/SSP.2003.1289387</a>.'
  mla: Schreier, Peter J., et al. “Canonical Coordinates Are the Right Coordinate
    System for Transform Coding of Noisy Sources.” <i>Proc.\ IEEE Works.\ Statistical
    Signal Proces.</i>, 2003, pp. 234–237, doi:<a href="https://doi.org/10.1109/SSP.2003.1289387">10.1109/SSP.2003.1289387</a>.
  short: 'P.J. Schreier, L.L. Scharf, T. Hu, S.D. Voran, in: Proc.\ IEEE Works.\ Statistical
    Signal Proces., 2003, pp. 234–237.'
date_created: 2023-01-30T11:52:09Z
date_updated: 2023-01-30T11:52:49Z
department:
- _id: '263'
doi: 10.1109/SSP.2003.1289387
page: 234–237
publication: Proc.\ IEEE Works.\ Statistical Signal Proces.
status: public
title: Canonical coordinates are the right coordinate system for transform coding
  of noisy sources
type: conference
user_id: '43497'
year: '2003'
...
---
_id: '40902'
abstract:
- lang: eng
  text: Recently, a number of papers have been published that show significant performance
    gains can be obtained by accounting for the fact that communication signals can
    be improper. In this paper, we derive a general result comparing the performance
    of conventional processing, which ignores the improper nature of signals, with
    processing that takes it into account. In particular, for an estimation and a
    detection problem, we find that the performance gain, as measured by mean-squared
    error and deflection, respectively, can be as large as a factor of 2, but no larger.
    In a communications example, we show how this finding generalizes the result that
    coherent processing enjoys a 3 dB gain over non-coherent processing.
author:
- first_name: Peter J.
  full_name: Schreier, Peter J.
  last_name: Schreier
- first_name: Louis L.
  full_name: Scharf, Louis L.
  last_name: Scharf
- first_name: Clifford T.
  full_name: Mullis, Clifford T.
  last_name: Mullis
citation:
  ama: 'Schreier PJ, Scharf LL, Mullis CT. A unified approach to performance comparisons
    between linear and widely linear processing. In: <i>Proc.\ IEEE Works.\ Statistical
    Signal Proces.</i> ; 2003:114–117. doi:<a href="https://doi.org/10.1109/SSP.2003.1289353">10.1109/SSP.2003.1289353</a>'
  apa: Schreier, P. J., Scharf, L. L., &#38; Mullis, C. T. (2003). A unified approach
    to performance comparisons between linear and widely linear processing. <i>Proc.\
    IEEE Works.\ Statistical Signal Proces.</i>, 114–117. <a href="https://doi.org/10.1109/SSP.2003.1289353">https://doi.org/10.1109/SSP.2003.1289353</a>
  bibtex: '@inproceedings{Schreier_Scharf_Mullis_2003, title={A unified approach to
    performance comparisons between linear and widely linear processing}, DOI={<a
    href="https://doi.org/10.1109/SSP.2003.1289353">10.1109/SSP.2003.1289353</a>},
    booktitle={Proc.\ IEEE Works.\ Statistical Signal Proces.}, author={Schreier,
    Peter J. and Scharf, Louis L. and Mullis, Clifford T.}, year={2003}, pages={114–117}
    }'
  chicago: Schreier, Peter J., Louis L. Scharf, and Clifford T. Mullis. “A Unified
    Approach to Performance Comparisons between Linear and Widely Linear Processing.”
    In <i>Proc.\ IEEE Works.\ Statistical Signal Proces.</i>, 114–117, 2003. <a href="https://doi.org/10.1109/SSP.2003.1289353">https://doi.org/10.1109/SSP.2003.1289353</a>.
  ieee: 'P. J. Schreier, L. L. Scharf, and C. T. Mullis, “A unified approach to performance
    comparisons between linear and widely linear processing,” in <i>Proc.\ IEEE Works.\
    Statistical Signal Proces.</i>, 2003, pp. 114–117, doi: <a href="https://doi.org/10.1109/SSP.2003.1289353">10.1109/SSP.2003.1289353</a>.'
  mla: Schreier, Peter J., et al. “A Unified Approach to Performance Comparisons between
    Linear and Widely Linear Processing.” <i>Proc.\ IEEE Works.\ Statistical Signal
    Proces.</i>, 2003, pp. 114–117, doi:<a href="https://doi.org/10.1109/SSP.2003.1289353">10.1109/SSP.2003.1289353</a>.
  short: 'P.J. Schreier, L.L. Scharf, C.T. Mullis, in: Proc.\ IEEE Works.\ Statistical
    Signal Proces., 2003, pp. 114–117.'
date_created: 2023-01-30T11:52:09Z
date_updated: 2023-01-30T11:52:52Z
department:
- _id: '263'
doi: 10.1109/SSP.2003.1289353
page: 114–117
publication: Proc.\ IEEE Works.\ Statistical Signal Proces.
status: public
title: A unified approach to performance comparisons between linear and widely linear
  processing
type: conference
user_id: '43497'
year: '2003'
...
---
_id: '40899'
abstract:
- lang: eng
  text: We challenge the perception that we live in a “proper world”, where complex
    random signals can always be assumed to be proper (also called circularly symmetric).
    Rather, we stress the fact that the analytic signal constructed from a nonstationary
    real signal is, in general, improper, which means that its complementary correlation
    function is nonzero. We explore the consequences of this finding in the context
    of stochastic time-frequency analysis in Cohen’s class. There, the analytic signal
    plays a prominent role because it reduces interference terms. However, the usual
    time-frequency representation (TFR) based on the analytic signal gives only an
    incomplete signal description. It must be augmented by a complementary TFR whose
    properties we develop in detail. We show why it is still advantageous to use the
    pair of standard and complementary TFRs of the analytic signal rather than the
    TFR of the corresponding real signal.
author:
- first_name: Peter J.
  full_name: Schreier, Peter J.
  last_name: Schreier
- first_name: Louis L.
  full_name: Scharf, Louis L.
  last_name: Scharf
citation:
  ama: 'Schreier PJ, Scharf LL. Stochastic time-frequency analysis using the analytic
    signal: why the complementary distribution matters. <i>{IEEE} {T}rans\ {S}ignal\
    {P}rocess</i>. 2003;51(12):3071–3079. doi:<a href="https://doi.org/10.1109/TSP.2003.818911">10.1109/TSP.2003.818911</a>'
  apa: 'Schreier, P. J., &#38; Scharf, L. L. (2003). Stochastic time-frequency analysis
    using the analytic signal: why the complementary distribution matters. <i>{IEEE}
    {T}rans.\ {S}ignal\ {P}rocess.</i>, <i>51</i>(12), 3071–3079. <a href="https://doi.org/10.1109/TSP.2003.818911">https://doi.org/10.1109/TSP.2003.818911</a>'
  bibtex: '@article{Schreier_Scharf_2003, title={Stochastic time-frequency analysis
    using the analytic signal: why the complementary distribution matters}, volume={51},
    DOI={<a href="https://doi.org/10.1109/TSP.2003.818911">10.1109/TSP.2003.818911</a>},
    number={12}, journal={{IEEE} {T}rans.\ {S}ignal\ {P}rocess.}, author={Schreier,
    Peter J. and Scharf, Louis L.}, year={2003}, pages={3071–3079} }'
  chicago: 'Schreier, Peter J., and Louis L. Scharf. “Stochastic Time-Frequency Analysis
    Using the Analytic Signal: Why the Complementary Distribution Matters.” <i>{IEEE}
    {T}rans.\ {S}ignal\ {P}rocess.</i> 51, no. 12 (2003): 3071–3079. <a href="https://doi.org/10.1109/TSP.2003.818911">https://doi.org/10.1109/TSP.2003.818911</a>.'
  ieee: 'P. J. Schreier and L. L. Scharf, “Stochastic time-frequency analysis using
    the analytic signal: why the complementary distribution matters,” <i>{IEEE} {T}rans.\
    {S}ignal\ {P}rocess.</i>, vol. 51, no. 12, pp. 3071–3079, 2003, doi: <a href="https://doi.org/10.1109/TSP.2003.818911">10.1109/TSP.2003.818911</a>.'
  mla: 'Schreier, Peter J., and Louis L. Scharf. “Stochastic Time-Frequency Analysis
    Using the Analytic Signal: Why the Complementary Distribution Matters.” <i>{IEEE}
    {T}rans.\ {S}ignal\ {P}rocess.</i>, vol. 51, no. 12, 2003, pp. 3071–3079, doi:<a
    href="https://doi.org/10.1109/TSP.2003.818911">10.1109/TSP.2003.818911</a>.'
  short: P.J. Schreier, L.L. Scharf, {IEEE} {T}rans.\ {S}ignal\ {P}rocess. 51 (2003)
    3071–3079.
date_created: 2023-01-30T11:52:08Z
date_updated: 2023-01-30T11:54:03Z
department:
- _id: '263'
doi: 10.1109/TSP.2003.818911
intvolume: '        51'
issue: '12'
page: 3071–3079
publication: '{IEEE} {T}rans.\ {S}ignal\ {P}rocess.'
status: public
title: 'Stochastic time-frequency analysis using the analytic signal: why the complementary
  distribution matters'
type: journal_article
user_id: '43497'
volume: 51
year: '2003'
...
---
_id: '40904'
abstract:
- lang: eng
  text: We present a comprehensive treatment of the second-order theory of complex
    random vectors and wide-sense stationary (WSS) signals. The main focus is on the
    improper case, in which the complementary covariance does not vanish. Accounting
    for the information present in the complementary covariance requires the use of
    widely linear transformations. Based on these, we present the eigenanalysis of
    complex vectors and apply it to the problem of rank reduction through principal
    components. We also investigate joint properties of two complex vectors by introducing
    canonical correlations, which paves the way for a discussion of the Wiener filter
    and its rank-reduced version. We link the concepts of propriety and joint propriety
    to eigenanalysis and canonical correlation analysis, respectively. Our treatment
    is extended to WSS signals. In particular, we give a result on the asymptotic
    distribution of eigenvalues and examine the connection between WSS, proper, and
    analytic signals.
author:
- first_name: Peter J.
  full_name: Schreier, Peter J.
  last_name: Schreier
- first_name: Louis L.
  full_name: Scharf, Louis L.
  last_name: Scharf
citation:
  ama: Schreier PJ, Scharf LL. Second-order analysis of improper complex random vectors
    and processes. <i>{IEEE} {T}rans\ {S}ignal\ {P}rocess</i>. 2003;51(3):714–725.
    doi:<a href="https://doi.org/10.1109/TSP.2002.808085">10.1109/TSP.2002.808085</a>
  apa: Schreier, P. J., &#38; Scharf, L. L. (2003). Second-order analysis of improper
    complex random vectors and processes. <i>{IEEE} {T}rans.\ {S}ignal\ {P}rocess.</i>,
    <i>51</i>(3), 714–725. <a href="https://doi.org/10.1109/TSP.2002.808085">https://doi.org/10.1109/TSP.2002.808085</a>
  bibtex: '@article{Schreier_Scharf_2003, title={Second-order analysis of improper
    complex random vectors and processes}, volume={51}, DOI={<a href="https://doi.org/10.1109/TSP.2002.808085">10.1109/TSP.2002.808085</a>},
    number={3}, journal={{IEEE} {T}rans.\ {S}ignal\ {P}rocess.}, author={Schreier,
    Peter J. and Scharf, Louis L.}, year={2003}, pages={714–725} }'
  chicago: 'Schreier, Peter J., and Louis L. Scharf. “Second-Order Analysis of Improper
    Complex Random Vectors and Processes.” <i>{IEEE} {T}rans.\ {S}ignal\ {P}rocess.</i>
    51, no. 3 (2003): 714–725. <a href="https://doi.org/10.1109/TSP.2002.808085">https://doi.org/10.1109/TSP.2002.808085</a>.'
  ieee: 'P. J. Schreier and L. L. Scharf, “Second-order analysis of improper complex
    random vectors and processes,” <i>{IEEE} {T}rans.\ {S}ignal\ {P}rocess.</i>, vol.
    51, no. 3, pp. 714–725, 2003, doi: <a href="https://doi.org/10.1109/TSP.2002.808085">10.1109/TSP.2002.808085</a>.'
  mla: Schreier, Peter J., and Louis L. Scharf. “Second-Order Analysis of Improper
    Complex Random Vectors and Processes.” <i>{IEEE} {T}rans.\ {S}ignal\ {P}rocess.</i>,
    vol. 51, no. 3, 2003, pp. 714–725, doi:<a href="https://doi.org/10.1109/TSP.2002.808085">10.1109/TSP.2002.808085</a>.
  short: P.J. Schreier, L.L. Scharf, {IEEE} {T}rans.\ {S}ignal\ {P}rocess. 51 (2003)
    714–725.
date_created: 2023-01-30T11:52:09Z
date_updated: 2023-01-30T11:52:55Z
department:
- _id: '263'
doi: 10.1109/TSP.2002.808085
intvolume: '        51'
issue: '3'
page: 714–725
publication: '{IEEE} {T}rans.\ {S}ignal\ {P}rocess.'
status: public
title: Second-order analysis of improper complex random vectors and processes
type: journal_article
user_id: '43497'
volume: 51
year: '2003'
...
---
_id: '40905'
abstract:
- lang: eng
  text: The analytic signal is commonly used in stochastic time-frequency analysis
    in Cohen’s class to reduce interference terms. However, we show that the usual
    time-frequency representation (TFR) based on the analytic signal gives only an
    incomplete signal description. This is because the analytic signal constructed
    from a non-stationary real signal is in general improper, which means that it
    has non-zero complementary correlation. We show how to augment the standard TFR
    by a complementary TFR to obtain a complete second-order characterization of the
    signal while still reducing interference terms compared to the TFR of the real
    signal.
author:
- first_name: Peter J.
  full_name: Schreier, Peter J.
  last_name: Schreier
- first_name: Louis L.
  full_name: Scharf, Louis L.
  last_name: Scharf
citation:
  ama: 'Schreier PJ, Scharf LL. Reducing interference in stochastic time-frequency
    analysis without losing information. In: <i>Proc. 36th\ Asilomar Conf.\ Signals
    Syst.\ Computers</i>. Vol 2. ; 2002:1565–1570. doi:<a href="https://doi.org/10.1109/ACSSC.2002.1197041">10.1109/ACSSC.2002.1197041</a>'
  apa: Schreier, P. J., &#38; Scharf, L. L. (2002). Reducing interference in stochastic
    time-frequency analysis without losing information. <i>Proc. 36th\ Asilomar Conf.\
    Signals Syst.\ Computers</i>, <i>2</i>, 1565–1570. <a href="https://doi.org/10.1109/ACSSC.2002.1197041">https://doi.org/10.1109/ACSSC.2002.1197041</a>
  bibtex: '@inproceedings{Schreier_Scharf_2002, title={Reducing interference in stochastic
    time-frequency analysis without losing information}, volume={2}, DOI={<a href="https://doi.org/10.1109/ACSSC.2002.1197041">10.1109/ACSSC.2002.1197041</a>},
    booktitle={Proc. 36th\ Asilomar Conf.\ Signals Syst.\ Computers}, author={Schreier,
    Peter J. and Scharf, Louis L.}, year={2002}, pages={1565–1570} }'
  chicago: Schreier, Peter J., and Louis L. Scharf. “Reducing Interference in Stochastic
    Time-Frequency Analysis without Losing Information.” In <i>Proc. 36th\ Asilomar
    Conf.\ Signals Syst.\ Computers</i>, 2:1565–1570, 2002. <a href="https://doi.org/10.1109/ACSSC.2002.1197041">https://doi.org/10.1109/ACSSC.2002.1197041</a>.
  ieee: 'P. J. Schreier and L. L. Scharf, “Reducing interference in stochastic time-frequency
    analysis without losing information,” in <i>Proc. 36th\ Asilomar Conf.\ Signals
    Syst.\ Computers</i>, 2002, vol. 2, pp. 1565–1570, doi: <a href="https://doi.org/10.1109/ACSSC.2002.1197041">10.1109/ACSSC.2002.1197041</a>.'
  mla: Schreier, Peter J., and Louis L. Scharf. “Reducing Interference in Stochastic
    Time-Frequency Analysis without Losing Information.” <i>Proc. 36th\ Asilomar Conf.\
    Signals Syst.\ Computers</i>, vol. 2, 2002, pp. 1565–1570, doi:<a href="https://doi.org/10.1109/ACSSC.2002.1197041">10.1109/ACSSC.2002.1197041</a>.
  short: 'P.J. Schreier, L.L. Scharf, in: Proc. 36th\ Asilomar Conf.\ Signals Syst.\
    Computers, 2002, pp. 1565–1570.'
date_created: 2023-01-30T11:52:09Z
date_updated: 2023-01-30T11:52:58Z
department:
- _id: '263'
doi: 10.1109/ACSSC.2002.1197041
intvolume: '         2'
page: 1565–1570
publication: Proc. 36th\ Asilomar Conf.\ Signals Syst.\ Computers
status: public
title: Reducing interference in stochastic time-frequency analysis without losing
  information
type: conference
user_id: '43497'
volume: 2
year: '2002'
...
---
_id: '40906'
abstract:
- lang: eng
  text: 'We consider the problem of minimum mean squared error (MMSE) estimation of
    complex random vectors in the improper case. Accounting for the information present
    in the complementary covariance requires the use of widely linear transformations.
    Based on these, we present the eigenanalysis of improper complex random vectors.
    This paves the way for a study of two different rank-reduced implementations of
    the complex Wiener Filter that make use of canonical coordinates: one that is
    optimum with respect to maximizing mutual information and one that minimizes mean
    squared error.'
author:
- first_name: Peter J.
  full_name: Schreier, Peter J.
  last_name: Schreier
- first_name: Louis L.
  full_name: Scharf, Louis L.
  last_name: Scharf
citation:
  ama: 'Schreier PJ, Scharf LL. Canonical coordinates for reduced-rank estimation
    of improper complex random vectors. In: <i>Proc.\ IEEE Int.\ Conf.\ Acoustics,
    Speech and Signal Process.</i> Vol 2. ; 2002:1153–1156. doi:<a href="https://doi.org/10.1109/ICASSP.2002.5744004">10.1109/ICASSP.2002.5744004</a>'
  apa: Schreier, P. J., &#38; Scharf, L. L. (2002). Canonical coordinates for reduced-rank
    estimation of improper complex random vectors. <i>Proc.\ IEEE Int.\ Conf.\ Acoustics,
    Speech and Signal Process.</i>, <i>2</i>, 1153–1156. <a href="https://doi.org/10.1109/ICASSP.2002.5744004">https://doi.org/10.1109/ICASSP.2002.5744004</a>
  bibtex: '@inproceedings{Schreier_Scharf_2002, title={Canonical coordinates for reduced-rank
    estimation of improper complex random vectors}, volume={2}, DOI={<a href="https://doi.org/10.1109/ICASSP.2002.5744004">10.1109/ICASSP.2002.5744004</a>},
    booktitle={Proc.\ IEEE Int.\ Conf.\ Acoustics, Speech and Signal Process.}, author={Schreier,
    Peter J. and Scharf, Louis L.}, year={2002}, pages={1153–1156} }'
  chicago: Schreier, Peter J., and Louis L. Scharf. “Canonical Coordinates for Reduced-Rank
    Estimation of Improper Complex Random Vectors.” In <i>Proc.\ IEEE Int.\ Conf.\
    Acoustics, Speech and Signal Process.</i>, 2:1153–1156, 2002. <a href="https://doi.org/10.1109/ICASSP.2002.5744004">https://doi.org/10.1109/ICASSP.2002.5744004</a>.
  ieee: 'P. J. Schreier and L. L. Scharf, “Canonical coordinates for reduced-rank
    estimation of improper complex random vectors,” in <i>Proc.\ IEEE Int.\ Conf.\
    Acoustics, Speech and Signal Process.</i>, 2002, vol. 2, pp. 1153–1156, doi: <a
    href="https://doi.org/10.1109/ICASSP.2002.5744004">10.1109/ICASSP.2002.5744004</a>.'
  mla: Schreier, Peter J., and Louis L. Scharf. “Canonical Coordinates for Reduced-Rank
    Estimation of Improper Complex Random Vectors.” <i>Proc.\ IEEE Int.\ Conf.\ Acoustics,
    Speech and Signal Process.</i>, vol. 2, 2002, pp. 1153–1156, doi:<a href="https://doi.org/10.1109/ICASSP.2002.5744004">10.1109/ICASSP.2002.5744004</a>.
  short: 'P.J. Schreier, L.L. Scharf, in: Proc.\ IEEE Int.\ Conf.\ Acoustics, Speech
    and Signal Process., 2002, pp. 1153–1156.'
date_created: 2023-01-30T11:52:09Z
date_updated: 2023-01-30T11:53:01Z
department:
- _id: '263'
doi: 10.1109/ICASSP.2002.5744004
intvolume: '         2'
page: 1153–1156
publication: Proc.\ IEEE Int.\ Conf.\ Acoustics, Speech and Signal Process.
status: public
title: Canonical coordinates for reduced-rank estimation of improper complex random
  vectors
type: conference
user_id: '43497'
volume: 2
year: '2002'
...
---
_id: '40907'
abstract:
- lang: eng
  text: In reduced-rank signal processing for radar, sonar, and digital communications,
    we seek the right tradeoff between model bias and model variance for reconstructing
    signals from noisy data. Here, we extend the classical theory by considering the
    low-rank approximation of complex random vectors, which may or may not be proper.
    We show that, in general, widely linear approximation is superior to strictly
    linear approximation, unless the vector to be approximated is proper, in which
    case the optimum procedure is strictly linear. We analyze the case where the approximated
    random vector becomes proper in its internal coordinate system. This class of
    random vector, which we call generalized proper, possesses qualities similar to
    proper random vectors.
author:
- first_name: Peter J.
  full_name: Schreier, Peter J.
  last_name: Schreier
- first_name: Louis L.
  full_name: Scharf, Louis L.
  last_name: Scharf
citation:
  ama: 'Schreier PJ, Scharf LL. Low-rank approximation of improper complex random
    vectors. In: <i>Proc. 35th\ Asilomar Conf.\ Signals Syst.\ Computers</i>. Vol
    1. ; 2001:597–601. doi:<a href="https://doi.org/10.1109/ACSSC.2001.986993">10.1109/ACSSC.2001.986993</a>'
  apa: Schreier, P. J., &#38; Scharf, L. L. (2001). Low-rank approximation of improper
    complex random vectors. <i>Proc. 35th\ Asilomar Conf.\ Signals Syst.\ Computers</i>,
    <i>1</i>, 597–601. <a href="https://doi.org/10.1109/ACSSC.2001.986993">https://doi.org/10.1109/ACSSC.2001.986993</a>
  bibtex: '@inproceedings{Schreier_Scharf_2001, title={Low-rank approximation of improper
    complex random vectors}, volume={1}, DOI={<a href="https://doi.org/10.1109/ACSSC.2001.986993">10.1109/ACSSC.2001.986993</a>},
    booktitle={Proc. 35th\ Asilomar Conf.\ Signals Syst.\ Computers}, author={Schreier,
    Peter J. and Scharf, Louis L.}, year={2001}, pages={597–601} }'
  chicago: Schreier, Peter J., and Louis L. Scharf. “Low-Rank Approximation of Improper
    Complex Random Vectors.” In <i>Proc. 35th\ Asilomar Conf.\ Signals Syst.\ Computers</i>,
    1:597–601, 2001. <a href="https://doi.org/10.1109/ACSSC.2001.986993">https://doi.org/10.1109/ACSSC.2001.986993</a>.
  ieee: 'P. J. Schreier and L. L. Scharf, “Low-rank approximation of improper complex
    random vectors,” in <i>Proc. 35th\ Asilomar Conf.\ Signals Syst.\ Computers</i>,
    2001, vol. 1, pp. 597–601, doi: <a href="https://doi.org/10.1109/ACSSC.2001.986993">10.1109/ACSSC.2001.986993</a>.'
  mla: Schreier, Peter J., and Louis L. Scharf. “Low-Rank Approximation of Improper
    Complex Random Vectors.” <i>Proc. 35th\ Asilomar Conf.\ Signals Syst.\ Computers</i>,
    vol. 1, 2001, pp. 597–601, doi:<a href="https://doi.org/10.1109/ACSSC.2001.986993">10.1109/ACSSC.2001.986993</a>.
  short: 'P.J. Schreier, L.L. Scharf, in: Proc. 35th\ Asilomar Conf.\ Signals Syst.\
    Computers, 2001, pp. 597–601.'
date_created: 2023-01-30T11:52:10Z
date_updated: 2023-01-30T11:52:35Z
department:
- _id: '263'
doi: 10.1109/ACSSC.2001.986993
intvolume: '         1'
page: 597–601
publication: Proc. 35th\ Asilomar Conf.\ Signals Syst.\ Computers
status: public
title: Low-rank approximation of improper complex random vectors
type: conference
user_id: '43497'
volume: 1
year: '2001'
...
---
_id: '40908'
abstract:
- lang: eng
  text: Block codes for use in turbo coding schemes provide an alternative to punctured
    convolutional codes when high rate component codes are needed. Since block codes
    have large, time-varying trellis diagrams, full maximum a posteriori (MAP) soft-in
    soft-out decoders are very complex. It is shown how to modify the MAP algorithm
    to utilize a sectionalized trellis diagram of the dual code for decoding, which
    minimizes computational complexity for high rate component codes. This paper also
    gives simulation results for some high rate block turbo codes
author:
- first_name: Peter J.
  full_name: Schreier, Peter J.
  last_name: Schreier
- first_name: Daniel J.
  full_name: Costello, Jr., Daniel J.
  last_name: Costello, Jr.
citation:
  ama: 'Schreier PJ, Costello, Jr. DJ. MAP decoding of linear block codes based on
    a sectionalized trellis of the dual code. In: <i>Proc.\ Int.\ Zurich Seminar Broadband
    Comm.</i> ; 2000:271–278. doi:<a href="https://doi.org/10.1109/IZSBC.2000.829262">10.1109/IZSBC.2000.829262</a>'
  apa: Schreier, P. J., &#38; Costello, Jr., D. J. (2000). MAP decoding of linear
    block codes based on a sectionalized trellis of the dual code. <i>Proc.\ Int.\
    Zurich Seminar Broadband Comm.</i>, 271–278. <a href="https://doi.org/10.1109/IZSBC.2000.829262">https://doi.org/10.1109/IZSBC.2000.829262</a>
  bibtex: '@inproceedings{Schreier_Costello, Jr._2000, title={MAP decoding of linear
    block codes based on a sectionalized trellis of the dual code}, DOI={<a href="https://doi.org/10.1109/IZSBC.2000.829262">10.1109/IZSBC.2000.829262</a>},
    booktitle={Proc.\ Int.\ Zurich Seminar Broadband Comm.}, author={Schreier, Peter
    J. and Costello, Jr., Daniel J.}, year={2000}, pages={271–278} }'
  chicago: Schreier, Peter J., and Daniel J. Costello, Jr. “MAP Decoding of Linear
    Block Codes Based on a Sectionalized Trellis of the Dual Code.” In <i>Proc.\ Int.\
    Zurich Seminar Broadband Comm.</i>, 271–278, 2000. <a href="https://doi.org/10.1109/IZSBC.2000.829262">https://doi.org/10.1109/IZSBC.2000.829262</a>.
  ieee: 'P. J. Schreier and D. J. Costello, Jr., “MAP decoding of linear block codes
    based on a sectionalized trellis of the dual code,” in <i>Proc.\ Int.\ Zurich
    Seminar Broadband Comm.</i>, 2000, pp. 271–278, doi: <a href="https://doi.org/10.1109/IZSBC.2000.829262">10.1109/IZSBC.2000.829262</a>.'
  mla: Schreier, Peter J., and Daniel J. Costello, Jr. “MAP Decoding of Linear Block
    Codes Based on a Sectionalized Trellis of the Dual Code.” <i>Proc.\ Int.\ Zurich
    Seminar Broadband Comm.</i>, 2000, pp. 271–278, doi:<a href="https://doi.org/10.1109/IZSBC.2000.829262">10.1109/IZSBC.2000.829262</a>.
  short: 'P.J. Schreier, D.J. Costello, Jr., in: Proc.\ Int.\ Zurich Seminar Broadband
    Comm., 2000, pp. 271–278.'
date_created: 2023-01-30T11:52:10Z
date_updated: 2023-01-30T11:52:39Z
department:
- _id: '263'
doi: 10.1109/IZSBC.2000.829262
page: 271–278
publication: Proc.\ Int.\ Zurich Seminar Broadband Comm.
status: public
title: MAP decoding of linear block codes based on a sectionalized trellis of the
  dual code
type: conference
user_id: '43497'
year: '2000'
...